What is the integral of (x^3+2x^2+5x+3)/(x^2+x+2) ?

The integral of (x^3+2x^2+5x+3)/(x^2+x+2) is

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integral of (x^3+2x^2+5x+3)/(x^2+x+2)

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} d x

\frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + C

Solution steps

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} d x

Expand

\frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} : \frac{x ^{3}}{x ^{2} + x + 2} + \frac{2 x ^{2}}{x ^{2} + x + 2} + \frac{5 x}{x ^{2} + x + 2} + \frac{3}{x ^{2} + x + 2}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x ^{2} + x + 2} d x + \int \frac{2 x ^{2}}{x ^{2} + x + 2} d x + \int \frac{5 x}{x ^{2} + x + 2} d x + \int \frac{3}{x ^{2} + x + 2} d x

\int \frac{x ^{3}}{x ^{2} + x + 2} d x = \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7})

\int \frac{2 x ^{2}}{x ^{2} + x + 2} d x = - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x

\int \frac{5 x}{x ^{2} + x + 2} d x = 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7}))

\int \frac{3}{x ^{2} + x + 2} d x = \frac{6}{7} arctan (\frac{2 x + 1}{7})

= \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x + 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{6}{7} arctan (\frac{2 x + 1}{7})

Simplify

\frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x + 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{6}{7} arctan (\frac{2 x + 1}{7}) : \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1))

= \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1))

Add a constant to the solution

= \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + C

\int \frac{2 x ^{3} - 7 x ^{2} + 3 x - 2}{( x ^{2} - x ) ^{2}} d x

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\int \frac{9}{( x + 1 ) ( x - 2 ) ^{2}} d x

\int (\frac{2 x - 1}{x ^{2} - x + 8}) d x

\int 7 x^{6} + 6 x^{2} + 2 x d x

What is the integral of (x^3+2x^2+5x+3)/(x^2+x+2) ?
The integral of (x^3+2x^2+5x+3)/(x^2+x+2) is